Mathematics
The Implicit Function Theorem and Its Applications
Quick fact
The implicit function theorem guarantees that even if an equation like x³ + y³ = 6xy can't be solved for y directly, near most points a unique differentiable function y = f(x) still exists. This is why we can differentiate such 'impossible' equations using implicit differentiation.
Why this is interesting
You've probably solved equations like y = x² + 3x for y. But what if an equation like x² + y² = 1 doesn't neatly give y by itself? How can we still find its slope at any point?